Snakes, Strings, Balloons and Other Active Contour Models

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Snakes, Strings, Balloons and Other Active Contour Models

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Snakes, Strings, Balloons and Other Active Contour Models

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Snakes, Strings, Balloonsand Other Active Contour Models

- Start with image and initial closed curve
- Evolve curve to lie along “important” features
- Edges
- Corners
- Detected features
- User input

- Region selection in Photoshop
- Segmentation of medical images
- Tracking

[Davatzikos and Prince]

[Davatzikos and Prince]

- Initialization: user-specified, automatic
- Curve properties: continuity, smoothness
- Image features: intensity, edges, corners, …
- Other forces: hard constraints, springs, attractors, repulsors, …
- Scale: local, multiresolution, global

- Framework: energy minimization, forces acting on curve
- Curve representation: ideal curve, sampled, spline, implicit function
- Evolution method: calculus of variations, numerical differential equations, local search

- Introduced by Kass, Witkin, and Terzopoulos
- Framework: energy minimization
- Bending and stretching curve = more energy
- Good features = less energy
- Curve evolves to minimize energy

- Also “Deformable Contours”

- Parametric representation of curve
- Energy functional consists of three terms

- First term is “membrane” term – minimum energy when curve minimizes length(“soap bubble”)
- Second term is “thin plate” term – minimum energy when curve is smooth

- Control a and b to vary between extremes
- Set b to 0 at a point to allow corner
- Set b to 0 everywhere to let curve follow sharp creases – “strings”

- Variety of terms give different effects
- For example,minimizes energy at intensity Idesired

- Gradient-based:
- Laplacian-based:
- In both cases, can smooth with Gaussian

- Can use corner detector we saw last time
- Alternatively, let q = tan-1 Iy / Ixand let nbe a unit vector perpendicular to the gradient. Then

- Spring
- Repulsion

- Computing forces on v that locally minimize energy gives differential equation for v
- Euler-Lagrange formula

- Discretize v: samples (xi, yi)
- Approximate derivatives with finite differences

- Iterative numerical solver

- Exact solution: calculus of variations
- Write equations directly in terms of forces,not energy
- Implicit equation solver
- Search neighborhood of each (xi, yi) for pixel that minimizes energy
- Shah & Williams paper

- Balloons [Cohen 91]
- Add inflation force
- Helps avoid getting stuck on small features

Balloons

Snakes

[Cohen 91]

[Cohen 91]

- Results of previously-run local algorithms
- e.g., Canny edge detector output convolvedwith Gaussian

- Automatically-evolved control points
- Others…

Add energy term for constant-color regions of a single color

Davatzikos and Prince

Davatzikos and Prince

Find features

and add

constraints

Davatzikos and Prince

Davatzikos and Prince

- In the simplest snakes algorithm, image features only attract locally
- Greater region of attraction: smooth image
- Curve might not follow high-frequency detail

- Multiresolution processing
- Start with smoothed image to attract curve
- Finish with unsmoothed image to get details

- Heuristic for global minimum vs. local minima

- Another way to attract curve to localized features: vector flow or diffusion methods
- Example:
- Find edges using Canny
- For each point, compute distance tonearest edge
- Push curve along gradient of distance field

Xu and Prince

Simple Snake

With Gradient Vector Field

Xu and Prince

Xu and Prince